We analyze the application to elastodynamic problems of mixed finite element
methods for elasticity with weak symmetry. Our approach leads to a semidiscrete
method which consists of a system of ordinary differential equations without
algebraic constraints. Our error analysis, which is based on a new elliptic
projection operator, applies to several mixed finite element spaces developed
for elastostatics. The error estimates we obtain are robust for nearly
incompressible materials.

We present stable mixed finite elements for planar linear
elasticity on general quadrilateral meshes. The symmetry of the
stress tensor is imposed weakly and so there are three primary
variables, the stress tensor, the displacement vector field,
and the scalar rotation. We develop and analyze a stable family
of methods, indexed by an integer r > 1 and with rate of
convergence in the L2 norm of order r for all the variables.
The methods use Raviart-Thomas elements for the stress, piecewise
tensor product polynomials for the displacement, and piecewise
polynomials for the rotation. We also present a simple first
order element, not belonging to this family. It uses the lowest
order BDM elements for the stress, and piecewise constants for the
displacement and rotation, and achieves first order convergence
for all three variables.

We study the approximation properties of a wide class of finite element differential forms
on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element
is the image of a cubical reference element under a diffeomorphism, and finite element
spaces in which the shape functions and degrees of freedom are obtained from the reference
element by pullback of differential forms. In the case where the diffeomorphisms from
the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard
that the rate of convergence in L2 exceeds by one the degree of the largest full polynomial
space contained in the reference space of shape functions. When the diffeomorphism is
multilinear, the rate of convergence for the same space of reference shape function
may degrade severely, the more so when the form degree is larger. The main result of
the paper gives a sufficient condition on the reference shape functions to obtain a given
rate of convergence.

We develop a family of finite element spaces of differential forms
defined on cubical meshes in any number of dimensions. The family
contains elements of all polynomial degrees and all form degrees.
In two dimensions, these include the serendipity finite elements and the
rectangular BDM elements. In three dimensions they include a recent
generalization of the serendipity spaces, and new H(curl) and H(div)
finite element spaces. Spaces in the family can be combined to give
finite element subcomplexes of the de Rham complex which satisfy the basic
hypotheses of the finite element exterior calculus, and hence can be used
for stable discretization of a variety of problems. The construction
and properties of the spaces are established in a uniform manner using
finite element exterior calculus.

In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial
codifferential operator on cochains, and they raised the question whether it is
consistent in the sense that for a smooth enough differential form the
combinatorial codifferential of the associated cochain converges to the
exterior codifferential of the form as the triangulation is refined. In 1991,
Smits proved this to be the case for the combinatorial codifferential applied
to 1-forms in two dimensions under the additional assumption that the initial
triangulation is refined in a completely regular fashion, by dividing each
triangle into four similar triangles. In this paper we extend Smits's result to
arbitrary dimensions, showing that the combinatorial codifferential on 1-forms
is consistent if the triangulations are uniform or piecewise uniform in a
certain precise sense. We also show that this restriction on the triangulations
is needed, giving a counterexample in which a different regular refinement
procedure, namely Whitney's standard subdivision, is used. Further, we show by
numerical example that for 2-forms in three dimensions, the combinatorial
codifferential is not consistent even for the most regular subdivision process.

This paper presents a nonconforming finite element approximation of
the space of symmetric tensors with square integrable divergence,
on tetrahedral meshes. Used for stress approximation together with
the full space of piecewise linear vector fields for displacement,
this gives a stable mixed finite element method which is shown to be
linearly convergent for both the stress and displacement, and which is
significantly simpler than any stable conforming mixed finite element
method. The method may be viewed as the three-dimensional analogue
of a previously developed element in two dimensions. As in that case,
a variant of the method is proposed as well, in which the displacement
approximation is reduced to piecewise rigid motions and the stress space
is reduced accordingly, but the linear convergence is retained.

This expository article discusses the meaning of
consistency, stability, and convergence of numerical discretizations of differential
equations. It provides a general framework for quantifying them in which
the fundamental theorem that consistency and stability imply convergence
can be rigorously stated and proved. These concepts are
illustrated with examples coming from both finite difference
methods and finite element methods.

In this paper, we consider the extension of the finite element exterior
calculus from elliptic problems, in which the Hodge Laplacian is an
appropriate model problem, to parabolic problems, for which we take the
Hodge heat equation as our model problem. The numerical method we study
is a Galerkin method based on a mixed variational formulation and using
as subspaces the same spaces of finite element differential forms which
are used for elliptic problems. We analyze both the semidiscrete and
a fully-discrete numerical scheme.

We discuss the construction of ﬁnite element spaces of differential forms which satisfy
the crucial assumptions of the ﬁnite element exterior calculus, namely that they can be assembled
into subcomplexes of the de Rham complex which admit commuting projections. We present two
families of spaces in the case of simplicial meshes, and two other families in the case of cubical
meshes. We make use of the exterior calculus and the Koszul complex to deﬁne and understand the
spaces. These tools allow us to treat a wide variety of situations, which are often treated separately,
in a uniﬁed fashion.

We survey the reasons for the ongoing boycott of the publisher
Elsevier. We examine Elsevier's pricing and bundling policies,
restrictions on dissemination by authors, and lapses in ethics and peer
review, and we conclude with thoughts about the future of mathematical
publishing.

We consider the ﬁnite element solution of the vector Laplace
equation on a domain in two dimensions. For various choices of boundary
conditions, it is known that a mixed ﬁnite element method, in which
the rotation of the solution is introduced as a second unknown, is
advantageous, and appropriate choices of mixed ﬁnite element spaces
lead to a stable, optimally convergent discretization. However, the
theory that leads to these conclusions does not apply to the case of
Dirichlet boundary conditions, in which both components of the solution
vanish on the boundary. We show, by computational example, that indeed
such mixed ﬁnite elements do not perform optimally in this case, and
we analyze the suboptimal convergence that does occur. As we indicate,
these results have implications for the solution of the biharmonic
equation and of the Stokes equations using a mixed formulation involving
the vorticity.

We investigate the journal impact factor, focusing on the applied
mathematics category. We discuss impact factor manipulation and demonstrate
that the impact factor gives an inaccurate view of journal quality, which
is poorly correlated with expert opinion.

We give a new, simple, dimension-independent definition of the
serendipity finite element family. The shape functions are the
span of all monomials which are linear in at least s-r of the
variables where s is the degree of the monomial or, equivalently,
whose superlinear degree (total degree with respect to variables
entering at least quadratically) is at most r. The degrees
of freedom are given by moments of degree at most r-2d on each
face of dimension d. We establish unisolvence and a geometric
decomposition of the space.

This article, prepared in conjunction with Mathematics Awareness Month 2010 with
the theme of Mathematics and Sports, reviews several examples of how mathematics
eluciates physical phenomena pertaining to the golf drive. Specifically it discusses
the double-pendulum model of a golf swing, transfer of energy and momentum in the
club head/ball impact, and drag and lift in the flight of the golf ball.

This article reports on the confluence of two streams of research,
one emanating from the fields of numerical analysis and scientific
computation, the other from topology and geometry. In it we consider
the numerical discretization of partial differential equations that
are related to differential complexes so that de Rham cohomology
and Hodge theory are key tools for exploring the well-posedness of
the continuous problem. The discretization methods we consider are
finite element methods, in which a variational or weak formulation of
the PDE problem is approximated by restricting the trial subspace to
an appropriately constructed piecewise polynomial subspace. After a
brief introduction to finite element methods, we develop an abstract
Hilbert space framework for analyzing the stability and convergence of
such discretizations. In this framework, the differential complex is
represented by a complex of Hilbert spaces and stability is obtained by
transferring Hodge theoretic structures that ensure well-posedness of the
continuous problem from the continuous level to the discrete. We show
stable discretization discretization is achieved if the finite element
spaces satisfy two hypotheses: they can be arranged into a subcomplex
of this Hilbert complex, and there exists a bounded cochain projection
from that complex to the subcomplex. In the next part of the paper, we
consider the most canonical example of the abstract theory, in which the
Hilbert complex is the de Rham complex of a domain in Euclidean space.
We use the Koszul complex to construct two families of finite element
differential forms, show that these can be arranged in subcomplexes of
the de Rham complex in numerous ways, and for each construct a bounded
cochain projection. The abstract theory therefore applies to give the
stability and convergence of finite element approximations of the Hodge
Laplacian. Other applications are considered as well, especially the
elasticity complex and its application to the equations of elasticity.
Background material is included to make the presentation self-contained
for a variety of readers.

The stability properties of simple element choices for the mixed
formulation of the Laplacian are investigated numerically. The element
choices studied use vector Lagrange elements, i.e., the space of
continuous piecewise polynomials vector fields of degree at most r,
for the vector variable, and divergence of this space, which consists of
discontinuous piecewise polynomials of one degree lower, for the scalar
variable. For polynomial degrees r equal 2 or 3, this pair of spaces
was found to be stable for all mesh families tested. In particular,
it is stable on diagonal mesh families, in contrast to its behaviour
for the Stokes equations. For degree r equal 1, stability holds for
some meshes, but not for others. Additionally, convergence was observed
precisely for the methods that were observed to be stable. However,
it seems that optimal order L2 estimates for the vector variable,
known to hold for r>3, do not hold for lower degrees.

We study the two primary families of spaces of finite element
differential forms with respect to a simplicial mesh in any number of
space dimensions. These spaces are generalizations of the classical
finite element spaces for vector fields, frequently referred to as
Raviart-Thomas, Brezzi-Douglas-Marini, and Nédélec spaces. In the
present paper, we derive geometric decompositions of these spaces which
lead directly to explicit local bases for them, generalizing the Bernstein
basis for ordinary Lagrange finite elements. The approach applies to
both families of finite element spaces, for arbitrary polynomial degree,
arbitrary order of the differential forms, and an arbitrary simplicial
triangulation in any number of space dimensions. A prominent role in the
construction is played by the notion of a consistent family of extension
operators, which expresses in an abstract framework a sufficient condition
for deriving a geometric decomposition of a finite element space leading
to a local basis.

The authors' video, also titled Moebius Transformations
Revealed, presents a visualization of these complex functions.
It has attracted the attention of the general public as well as
a technical audience. Here, we explain the mathematics involved,
as well as how the video was produced.

We construct finite element subspaces of the space of symmetric tensors
with square-integrable divergence on a three-dimensional domain. These
spaces can be used to approximate the stress field in the classical
Hellinger-Reissner mixed formulation of the elasticty equations, when
standard discontinous finite element spaces are used to approximate
the displacement field. These finite element spaces are defined
with respect to an arbitrary simplicial triangulation of the domain,
and there is one for each positive value of the polynomial degree
used for the displacements. For each degree, these provide a stable
finite element discretization. The construction of the spaces is closely
tied to discretizations of the elasticity complex, and can be viewed as
the three-dimensional analogue of the triangular element family for plane
elasticity previously proposed by Arnold and Winther.

A differential form is a field which assigns to each point
of a domain an alternating multilinear form on its tangent space.
The exterior derivative operation, which maps differential forms to
differential forms of the next higher order, unifies the basic first order
differential operators of calculus, and is a building block for a great
variety of differential equations. When discretizing such differential
equations by finite element methods, stable discretization depends
on the development of spaces of finite element differential forms.
As revealed recently through the finite element exterior calculus,
for each order of differential form, there are two natural families
of finite element subspaces associated to a simplicial triangulation.
In the case of forms of order zero, which are simply functions, these two
families reduce to one, which is simply the well-known family of Lagrange
finite element subspaces of the first order Sobolev space. For forms
of degree 1 and of degree n-1 (where n is the space dimension),
we obtain two natural families of finite element subspaces, unifying many
of the known mixed finite element spaces developed over the last decades.

In this paper, we construct new finite element methods for the
approximation of the equations of linear elasticity in three space
dimensions that produce direct approximations to both stresses and
displacements. The methods are based on a modified form of the
Hellinger-Reissner variational principle that only weakly imposes the
symmetry condition on the stresses. Although this approach has been
previously used by a number of authors, a key new ingredient here is
a constructive derivation of the elasticity complex starting
from the de Rham complex. By mimicking this construction in the
discrete case, we derive new mixed finite elements for elasticity in a
systematic manner from known discretizations of the de Rham complex.
These elements appear to be simpler than the ones previously derived.
For example, we construct stable discretizations which use only
piecewise linear elements to approximate the stress field and piecewise
constant functions to approximate the displacement field.

Specifying boundary conditions continues to be a
challenge in numerical relativity in order to obtain a long time
convergent numerical simulation of Einstein's equations in domains with
artificial boundaries. In this paper, we address this problem for the
Einstein-Christoffel (EC) symmetric hyperbolic formulation of
Einstein's equations linearized around flat spacetime. First, we
prescribe simple boundary conditions that make the problem well posed
and preserve the constraints. Next, we indicate boundary conditions for
a system that extends the linearized EC system by including the
momentum constraints and whose solution solves Einstein's equations in
a bounded domain. Finally, we extend our results to the case of
inhomogeneous boundary conditions.

In a recent paper of Arnold, Brezzi, and Marini, the
ideas of discontinuous Galerkin methods were used to obtain and analyze
two new families of locking free finite element methods for the
approximation of the Reissner-Mindlin plate problem. By following
their basic approach, but making different choices of finite element
spaces, we develop and analyze other families of locking free finite
elements that eliminate the need for the introduction of a reduction
operator, which has been a central feature of many locking-free
methods. For k>1, all the methods use piecewise polynomials of
degree k to approximate the transverse displacement and (possibly
subsets) of piecewise polynomials of degree k-1 to approximate both
the rotation and shear stress vectors. The approximation spaces for
the rotation and the shear stress are always identical. The methods
vary in the amount of interelement continuity required. In terms of
smallest number of degrees of freedom, the simplest method approximates
the transverse displacement with continuous, piecewise quadratics and
both the rotation and shear stress with rotated linear
Brezzi-Douglas-Marini elements.

Finite element exterior calculus is an approach to the
design and understanding of finite element discretizations for a wide
variety of systems of partial differential equations. This approach
brings to bear tools from differential geometry, algebraic topology,
and homological algebra to develop discretizations which are compatible
with the geometric, topological, and algebraic structures which
underlie well-posedness of the PDE problem being solved. In the finite
element exterior calculus, many finite element spaces are revealed as
spaces of piecewise polynomial differential forms. These connect to
each other in discrete subcomplexes of elliptic differential complexes,
and are also related to the continuous elliptic complex through
projections which commute with the complex differential. Applications
are made to the finite element discretization of a variety of problems,
including the Hodge Laplacian, Maxwell's equations, the equations of
elasticity, and elliptic eigenvalue problems, and also to
preconditioners.

A close connection between the ordinary de Rham complex
and a corresponding elasticity complex is utilized to derive new mixed
finite element methods for linear elasticity. For a formulation with
weakly imposed symmetry, this approach leads to methods which are
simpler than those previously obtained. For example, we construct
stable discretizations which use only piecewise linear elements to
approximate the stress field and piecewise constant functions to
approximate the displacement field. We also discuss how the strongly
symmetric methods proposed in [8] can be derived in the present
framework. The method of construction works in both two and three space
dimensions, but for simplicity the discussion here is limited to the
two dimensional case.

We present a family of stable rectangular mixed finite
elements for plane elasticity. Each member of the family consists of a
space of piecewise polynomials discretizing the space of symmetric
tensors in which the stress field is sought, and another to discretize
the space of vector fields in which the displacement is sought. These
may be viewed as analogues in the case of rectangular meshes of mixed
finite elements recently proposed for triangular meshes. As for the
triangular case the elements are closely related to a discrete version
of the elasticity differential complex.

We develop a family of locking-free elements for the
Reissner-Mindlin plate using Discontinuous Galerkin techniques,
one for each odd degree, and prove optimal error estimates.
A second family uses conforming elements for the rotations
and nonconforming elements for the transverse displacement,
generalizing the element of Arnold and Falk to higher degree.

We consider the approximation properties of quadrilateral
finite element spaces of vector fields defined by the Piola transform,
extending results previously obtained for scalar approximation. The
finite element spaces are constructed starting with a given finite
dimensional space of vector fields on a square reference element, which
is then transformed to a space of vector fields on each convex
quadrilateral element via the Piola transform associated to a bilinear
isomorphism of the square onto the element. For affine isomorphisms, a
necessary and sufficientcondition for approximation of order r+1 in L2
is that each component ofthe given space of functions on the reference
element contain all polynomial functions of total degree at most r. In
the case of bilinear isomorphisms,the situation is more complicated and
we give a precise characterization of what is needed for optimal order
L2-approximation of the function and of its divergence. As
applications, we demonstrate degradation of the convergence order on
quadrilateral meshes as compared to rectangular meshes for some
standard finite element approximations of H(div). We also derive new
estimates for approximation by quadrilateral Raviart-Thomas elements
(requiring less regularity) and propose a new quadrilateral finite
element space which provides optimal order approximation in H(div).
Finally, we demonstrate the theory with numerical computations of mixed
and least squares finite element aproximations of the solution of
Poisson's equation.

We derive a new first-order formulation for Einstein's equations which
involves fewer unknowns than other first-order formulations that have
been proposed. The new formulation is based on the 3+1 decomposition
with arbitrary lapse and shift. In the reduction to first order form
only 8 particular combinations of the 18 first derivatives of the spatial
metric are introduced. In the case of linearization about Minkowski
space, the new formulation consists of symmetric hyperbolic system in
14 unknowns, namely the components of the extrinsic curvature
perturbation and the 8 new variables, from whose solution the metric
perturbation can be computed by integration.

In this paper we propose a way to analyze certain classes
of dimension reduction models for elliptic problems in thin domains.
We develop asymptotic expansions for the exact and model solutions,
having the thickness as small parameter. The modeling error is then
estimated by comparing the respective expansions, and the upper bounds
obtained make clear the influence of the order of the model and the
thickness on the convergence rates. The techniques developed here
allows for estimates in several norms and semi-norms, and also
interior estimates (which disregards boundary layers).

We present a family of pairs of finite element spaces for
the unaltered Hellinger-Reissner variational principle using
polynomial shape functions on a single triangular mesh for stress and
displacement. There is a member of the family for each polynomial
degree, beginning with degree two for the stress and degree one for the
displacement, and each is stable and affords optimal order
approximation. The simplest element pair involves 24 local degrees of
freedom for the stress and 6 for the displacement. We also construct a
lower order element involving 21 stress degrees of freedom and 3
displacement degrees of freedom which is, we believe, likely to be the
simplest possible conforming stable element pair with polynomial shape
functions. For all these conforming elements the approximate stress not
only belongs to H(div), but is also continuous at element vertices,
which is more continuity than may be desired. We show that for
conforming finite elements with polynomial shape functions, this
additional continuity is unavoidable. To overcome this obstruction, we
construct as well some non-conforming stable mixed finite elements,
which we show converge with optimal order as well. The simplest of
these involves only 12 stress and 6 displacement degrees of freedom on
each triangle.

We construct first order, stable, nonconforming mixed
finite elements for plane elasticity and analyze their convergence.
The mixed method is based on the Hellinger-Reissner variational
formulation in which the stress and displacement fields are the primary
unknowns. The stress elements use polynomial shape functions but do
not involve vertex degrees of freedom.

Differential complexes such as the de Rham
complex have recently come to play an important role in the design and
analysis of numerical methods for partial differential equations. The
design of stable discretizations of systems of partial differential
equations often hinges on capturing subtle aspects of the structure of
the system in the discretization. In many cases the differential
geometric structure captured by a differential complex has proven
to be a key element, and a discrete differential complex which
is appropriately related to the original complex is essential. This
new geometric viewpoint has provided a unifying understanding of a
variety of innovative numerical methods developed over recent decades
and pointed the way to stable discretizations of problems for which
none were previously known, and it appears likely to play an important
role in attacking some currently intractable problems in numerical PDE.

Over the last two decades, there has been an extensive effort
to devise and analyze finite elements schemes for the approximation of
the Reissner�Mindlin plate equations which avoid locking, numerical
overstiffness resulting in a loss of accuracy when the plate is thin.
There are now many triangular and rectangular finite elements, for
which a mathematical analysis exists to certify them as free of
locking. Generally speaking, the analysis for rectangular elements
extends to the case of parallograms, which are defined by affine
mappings of rectangles. However, for more general convex
quadrilaterals, defined by bilinear mappings of rectangles, the
analysis is more complicated. Recent results by the authors on the
approximation properties of quadrilateral finite elements shed some
light on the problems encountered. In particular, they show that for
some finite element methods for the approximation of the
Reissner-Mindlin plate, the obvious generalization of rectangular
elements to general quadrilateral meshes produce methods which lose
accuracy. In this paper, we present an overview of this situation.

We show that the Reissner-Mindlin plate bending model
has a wider range of applicability than the Kirchhoff-Love model
for the approximation of clamped linearly elastic plates.
Under the assumption that the body force density is constant
in the transverse direction, the Reissner-Mindlin model
solution converges to the three-dimensional linear elasticity solution
in the relative energy norm for the full range of surface loads.
However, for loads with a significant transverse shear effect,
the Kirchhoff-Love model fails.

We provide a framework for the analysis of a large class
of discontinuous methods for second-order elliptic problems. It allows
for the understanding and comparison of most of the discontinuous
Galerkin methods that have been proposed for the numerical treatment
of elliptic problems by diverse communities over three decades.

There have been many efforts, dating back four decades, to
develop stable mixed finite elements for the stress-displacement
formulation of the plane elasticity system. This requires the
development of a compatible pair of finite element spaces, one to
discretize the space of symmetric tensors in which the stress field is
sought, and one to discretize the space of vector fields in which the
displacement is sought. Although there are number of well-known mixed
finite element pairs known for the analogous problem involving vector
fields and scalar fields, the symmetry of the stress field is a
substantial additional difficulty, and the elements presented here are
the first ones using polynomial shape functions which are known to be
stable. We present a family of such pairs of finite element spaces, one
for each polynomial degree, beginning with degree two for the stress
and degree one for the displacement, and show stability and optimal
order approximation. We also analyze some obstructions to the
construction of such finite element spaces, which account for the
paucity of elements available.

We consider the approximation properties of finite element
spaces on quadrilateral meshes. The finite element spaces are
constructed starting with a given finite dimensional space of functions
on a square reference element, which is then transformed to a space
of functions on each convex quadrilateral element via a bilinear
isomorphism of the square onto the element. It is known that for affine
isomorphisms, a necessary and sufficient condition for approximation
of order r+1 in L2 and order r in H1 is that the given space of
functions on the reference element contain all polynomial functions
of total degree at most r. In the case of bilinear isomorphisms, it
is known that the same estimates hold if the function space contains
all polynomial functions of separate degree r. We show, by means of
a counterexample, that this latter condition is also necessary. As
applications we demonstrate degradation of the convergence order on
quadrilateral meshes as compared to rectangular meshes for serendipity
finite elements and for various mixed and nonconforming finite elements.

Quadrilateral finite elements are generally constructed by
starting from a given finite dimensional space of polynomials V^
on the unit reference square K^. The elements of V^ are
then transformed by using the bilinear isomorphisms F_K which map
K^ to each convex quadrilateral element K. It has been recently
proven that a necessary and sufficient condition for approximation of
order r+1 in L^2 and r in H^1 is that V^ contains the
space Q_r of all polynomial functions of degree r separately in
each variable. In this paper several numerical experiments are
presented which confirm the theory. The tests are taken from various
examples of applications: the Laplace operator, the Stokes problem and
an eigenvalue problem arising in fluid-structure interaction modeling.

The construction of gravitational wave observatories is one
of the greatest scientific efforts of our time. As a result, there is
presently a strong need to numerically simulate the emission of
gravitation radiation from massive astronomical events such as black
hole collisions. This entails the numerical solution of the Einstein
field equations. We briefly describe the field equations in their
natural setting, namely as statements about the geometry of space time.
Next we describe the complicated system that arises when the field
equations are recast as partial differential equations, and discuss
procedures for deriving from them a more tractable system consisting of
constraint equations to be satisfied by initial data and together with
evolution equations. We present some applications of modern finite
element technology to the solution of the constraint equations in order
to find initial data relevant to black hole collisions. We conclude
by enumerating some of the many computational challenges that remain.

We provide a common framework for the understanding,
comparison, and analysis of several discontinuous Galerkin methods that
have been proposed for the numerical treatment of elliptic problems.
This class includes the recently introduced methods of Bassi and Rebay
(together with the variants proposed by Brezzi, Manzini, Marini, Pietra
and Russo), the local discontinuous Galerkin methods of Cockburn and
Shu, and the method of Baumann and Oden. It also includes the
so-called interior penalty methods developed some time ago by Douglas
and Dupont, Wheeler, Baker, and Arnold among others.

We consider the solution of systems of linear algebraic
equations which arise from the finite element discretization of
variational problems posed in the Hilbert spaces H(div) and H(curl) in
three dimensions. We show that if appropriate finite element spaces and
appropriate additive or multiplicative Schwarz smoothers are used, then
the multigrid V-cycle is an efficient solver and preconditioner for the
discrete operator. All results are uniform with respect to the mesh
size, the number of mesh levels, and weights on the two terms in the
inner products.

We present an algorithm for the construction of locally
adapted conformal tetrahedral meshes. The algorithm is based on
bisection of tetrahedra. A new data structure is introduced, which
simplifies both the selection of the refinement edge of a tetrahedron
and the recursive refinement to conformity of a mesh once some
tetrahedra have been bisected. We prove that repeated application of
the algorithm leads to only finitely many tetrahedral shapes up to
similarity, and bound the amount of additional refinement that is needed
to achieve conformity. Numerical examples of the effectiveness of the
algorithm are presented.

An adaptive finite element algorithm for elliptic
boundary value problems in 3 is presented. The algorithm uses
linear finite elements, a-posteriori error estimators, a mesh
refinement scheme based on bisection of tetrahedra, and a multi-grid
solver. We show that the repeated bisection of an arbitrary tetrahedron
leads to only a finite number of dissimilar tetrahedra, and that the
recursive algorithm ensuring conformity of the meshes produced
terminates in a finite number of steps. A procedure for assigning
numbers to tetrahedra in a mesh based on a-posteriori error estimates,
indicating the degree of refinement of the tetrahedron, is also
presented. Numerical examples illustrating the effectiveness of the
algorithm are given.

According to the theory of general relativity, the relative
acceleration of masses generates gravitational radiation. Although
gravitational radiation has not yet been detected, it is believed that
extremely violent cosmic events, such as the collision of black holes,
should generate gravity waves of sufficient amplitude to detect on
earth. The massive Laser Interferometer Gravitational-Wave Observatory,
or LIGO, is now being constructed to detect gravity waves. Consequently
there is great interest in the computer simulation of black hole
collisions and similar events, based on the numerical solution of the
Einstein field equations. In this note we introduce the scientific,
mathematical, and computational problems and discuss the development of
a computer code to solve the initial data problem for colliding black
holes, a nonlinear elliptic boundary value problem posed in an unbounded
three dimensional domain which is a key step in solving the full field
equations. The code is based on finite elements, adaptive meshes, and
a multigrid solution process. Here we will particularly emphasize
the mathematical and algorithmic issues arising in the generation of
adaptive tetrahedral meshes.

In an earlier paper we constructed and analyzed a multigrid
preconditioner for the system of linear algebraic equations arising from
the finite element discretization of boundary value problems associated
to the differential operator I - grad div. In this paper we analyze the
procedure without assuming that the underlying domain is convex and show
that, also in this case, the preconditioner is spectrally equivalent to
the inverse of the discrete operator.

We consider iterative methods for the solution of linear
systems of equations arising from mixed finite element discretization of
the Reissner-Mindlin plate model. We show how to construct symmetric
positive definite block diagonal preconditioners for these indefinite
systems such that the resulting systems have spectral condition numbers
independent of both the mesh size h and the plate thickness t.

Summarizing the work of [1], we show how to construct
preconditioners using domain decomposition and multigrid techniques for
the system of linear algebraic equations which arises from the finite
element discretization of boundary value problems associated to the
differential operator I - grad div. These preconditioners are shown
to be spectrally equivalent to the inverse of the operator and thus
may be used to precondition iterative methods so that any given error
reduction may be achieved in a finite number of iterations independent
of the mesh discretization. We describe applications of these results
to the efficient solution of mixed and least squares finite element
approximations of elliptic boundary value problems.

We consider the derivation of two-dimensional models for the
bending and stretching of a thin three-dimensional linearly elastic
plate using variational methods. Specifically we consider restriction
of the trial space in two different forms of the Hellinger-Reissner
variational principle for 3-D elasticity to functions with a specified
polynomial dependence in the transverse direction. Using this approach
many different plate models are possible and we classify and investigate
the most important. We study in detail a method which leads naturally
not only to familiar plate models, but also to error bounds between the
plate solution and the full 3-D solution.

We consider iterative methods for the solution of the linear
system of equations arising from the mixed finite element discretization
of the Reissner-Mindlin plate model. We show how to construct a
symmetric positive definite block diagonal preconditioner such that the
resulting linear system has spectral condition number independent of
both the mesh size h and the plate thickness t. We further discuss how
this preconditioner may be implemented and then apply it to efficiently
solve this indefinite linear system. Although the mixed formulation
of the Reissner-Mindlin problem has a saddle-point structure common to
other mixed variational problems, the presence of the small parameter t
and the fact that the matrix in the upper left corner of the partition
is only positive semidefinite introduces new complications.

We consider the solution of the system of linear algebraic
equations which arises from the finite element discretization of
boundary value problems associated to the differential operator I - grad
div. The natural setting for such problems is in the Hilbert space
H(div) and the variational formulation is based on the inner product in
H(div). We show how to construct preconditioners for these equations
using both domain decomposition and multigrid techniques. These
preconditioners are shown to be spectrally equivalent to the inverse
of the operator. As a consequence, they may be used to precondition
iterative methods so that any given error reduction may be achieved
in a finite number of iterations, with the number independent of
the mesh discretization. We describe applications of these results
to the efficient solution of mixed and least squares finite element
approximations of elliptic boundary value problems.

An analysis is presented for a recently proposed finite
element method for the Reissner-Mindlin plate problem. The method is
based on the standard variational principle, uses nonconforming linear
elements to approximate the rotations and conforming linear elements to
approximate the transverse displacements, and avoids the usual "locking
problem" by interpolating the shear stress into a rotated space of
lowest order Raviart-Thomas elements. When the plate thickness t=O(h),
it is proved that the method gives optimal order error estimates uniform
in t. However, the analysis suggests and numerical calculations confirm
that the method can produce poor approximations for moderate sized
values of the plate thickness. Indeed, for t fixed, the method does not
converge as the mesh size h tends to zero.

Interior error estimates are obtained for a low order
finite element introduced by Arnold and Falk for the Reissner-Mindlin
plates. It is proved that the approximation error of the finite element
solution in the interior domain is bounded above by two parts: one
measures the local approximability of the exact solution by the finite
element space and the other the global approximability of the finite
element method. As an application, we show that for the soft simply
supported plate, the Arnold-Falk element still achieves an almost
optimal convergence rate in the energy norm away from the boundary
layer, even though optimal order convergence cannot hold globally due
to the boundary layer. Numerical results are given which support our
conclusion.

We briefly present the main idea of partial selective reduced
integration as developed in other works of the authors. The idea is quite
general and can be applied to a number of different situations, but we
concentrate on the case of the Naghdi shell model.

We propose a new family of finite element methods for
the Naghdi shell model, one method associated with each nonnegative
integer k. The methods are based on a nonstandard mixed formulation,
and the kth method employs triangular Lagrange finite elements of
degree k+2 augmented by bubble functions of degree k+3 for both the
displacement and rotation variables, and discontinuous piecewise
polynomials of degree k for the shear and membrane stresses. This
method can be implemented in terms of the displacement and rotation
variables alone, as the minimization of an altered energy functional
over the space mentioned. The alteration consists of the introduction
of a weighted local projection into part, but not all, of the shear and
membrane energy terms of the usual Naghdi energy. The relative error
in the method, measured in a norm which combines the H1 norm of the
displacement and rotation fields and an appropriate norm of the shear
and membrane stress fields, converges to zero with order k+1 uniformly
with respect to the shell thickness for smooth solutions, at least under
the assumption that certain geometrical coefficients in the Nagdhi model
are replaced by piecewise constants.

We investigate the structure of the solution of the
Reissner-Mindlin plate equations in its dependence on the plate
thickness for various boundary conditions, developing asymptotic
expansions in powers of the plate thickness for the main physical
quantities. These expansions are uniform up to the boundary for the
transverse displacement, but for other variables there is a boundary
layer, whose strength depends on the boundary conditions. We give
rigorous error bounds for the errors in the expansions in Sobolev norms
and make various applications.

Nonconforming piecewise linear finite elements for the
velocity field and piecewise constant elements for the pressure field
give a simple stable, optimal order approximation to the Stokes
equations, but are not stable for the equations of incompressible
elasticty, which differ from the Stokes equations only in that the
vector Laplace operator is replaced by the Lame operator. However,
we show that if we replace the divergence by the rotation, then the
nonconforming linear-constant element is stable both for the system
involving the Laplacian and for that involving the Lame operator.
Finally we discuss an application to the Reissner-Mindlin plate.

We study the finite element approximation of the stationary
Stokes equations in the velocity-pressure formulation using continuous
piecewise quadratic functions for velocity and discontinuous piecewise
linear functions for pressure. For some meshes this method is unstable,
even after spurious pressure modes are removed. For other meshes there
are spurious local pressure modes, but once they are removed the method
is stable, and in particular, the velocity converges with optimal order.
On yet other meshes there are no spurious pressure modes and the method
is stable and optimally convergent for both pressure and velocity.

Finite element methods for the Reissner-Mindlin plate theory
are discussed. Methods in which both the tranverse displacement and the
rotation are approximated by finite elements of low degree mostly suffer
from locking. However a number of related methods have been devised
recently which avoid locking effects. Although the finite element
spaces for both the rotation and transverse displacement contain little
more than piecewise linear functions, optimal order convergence holds
uniformly in the thickness. The main ideas leading to such methods are
reviewed and the relationships between various methods are clarified.

This paper treats the basic ideas of mixed finite element
methods at an introductory level. Although the viewpoint presented is
that of a mathematician, the paper is aimed at practitioners and the
mathematical prerequisites are kept to a minimum. A classification
of variational principles and of the corresponding weak formulations
and Galerkin methods-displacement, equilibrium, and mixed-is given
and illustrated through four significant examples. The advantages
and disadvantages of mixed methods are discussed. The concepts of
convergence, approximability, and stability and their interrelations are
developed, and a resume is given of the stability theory which governs
the performance of mixed methods. The paper concludes with a survey of
techniques that have been developed for the construction of stable mixed
methods and numerous examples of such methods.

The structure of the solution of the Reissner-Mindlin model
of a clamped plate is investigated, emphasizing its dependence on the
plate thickness. Asymptotic expansions in powers of the plate thickness
are developed for the main physical quantities and the boundary layer
is studied. Rigorous error bounds are given for the errors in the
expansions in Sobolev norms. As applications, new regularity results
for the solutions and new estimates for the difference between the
Reissner-Mindlin solution and the solution to the biharmonic equation
are derived. Boundary conditions for a clamped edge are considered for
most of the paper, and the very similar case of a hard simply-supported
plate is discussed briefly at the end.

We investigate the structure of the solution of the
Reissner-Mindlin plate equations in its dependence on the plate
thickness for various boundary conditions, developing asymptotic
expansions in powers of the plate thickness for the main physical
quantities. These expansions are uniform up to the boundary for the
transverse displacement, but for other variables there is a boundary
layer, whose strength depends on the boundary conditions. We give
rigorous error bounds for the errors in the expansions in Sobolev norms
and make various applications.

The single layer heat potential operator, K, arises in the
solution of initial-boundary value problems for the heat equation using
boundary integral methods. In this note we show that K maps a certain
anisotropic Sobolev space isomorphically onto its dual, and, moreover,
satisfies the coercivity inequality <K q,q> >=c|q|^2. We thereby
establish the well-posedness of the operator equation K q=f and provide
a basis for the analysis of the discretizations.

We present and analyze a simple finite element method for the
Mindlin-Reissner plate model in the primitive variables. Our method uses
nonconforming linear finite elements for the transverse displacement and
conforming linear finite elements enriched by bubbles for the rotation,
with the computation of the element stiffness matrix modified by the
inclusion of a simple elementwise averaging. We prove that the method
converges with optimal order uniformly with respect to thickness.

The Dirichlet problem for Laplace's equation is often solved
by means of the single layer potential representation, leading to a
Fredholm integral equation of the first kind with logarithmic kernel.
We propose to solve this integral equation using a Petrov-Galerkin
method with trigonometric polynomials as test functions and, as trial
functions, a span of delta distributions centered at boundary points.
The approximate solution to the boundary value problem thus computed
converges exponentially away from the boundary and algebraically up to
the boundary. We show that these convergence results hold even when
the discretization matrices are computed via numerical quadratures.
Finally, we discuss our implementation of this method using the fast
Fourier transform to compute the discretization matrices, and present
numerical experiments in order to confirm our theory and to examine the
behavior of the method in cases where the theory doesn't apply due to
lack of smoothness.

We consider the existence of regular solutions to the
boundary value problem div U = f on a plane polygonal domain with the
Dirichlet boundary condition U=g. We formulate simultaneously necessary
and sufficient conditions on f and g in order that a solution U exist in
the Sobolev space W^s+1_p. In addition to the obvious regularity and
integral conditions these consist of at most one compatibility condition
at each vertex of the polygon. In the special case of homogeneous
boundary data, it is necessary and sufficient that f belong to W^s_p,
have mean value zero, and vanish at each vertex. (The latter condition
only applies if s is large enough that the point values make sense.)
We construct a solution operator which is independent of s and p. As
intermediate results we obtain various new trace theorems for Sobolev
spaces on polygons.

We propose a new mixed variational formulation for the
equations of linear elasticity. It does not require symmetric tensors
and consequently is easy to discretize by adapting mixed finite elements
developed for scalar second order elliptic equations.

The solution of the heat equation with Dirichlet boundary conditions
by the boundary integral method leads to an integral equation of the
first kind to determine the boundary flux. We show that the
linear operator so defined is an automorphism from a certain function
to another defined on the boundary and is coercive, thereby establishing
the well-posedness of the method.

We consider the equations of linear homogeneous anisotropic
elasticity admitting the possibility that the material is internally
constrained, and formulate a simple necessary and sufficient condition
for the fundamental boundary value problems to be well-posed. For
materials fulfilling the condition, we establish continuous dependence
of the displacement and stress on the elastic moduli and ellipticity of
the elasticity system. As an application we determine the orthotropic
materials for which the fundamental problems are well-posed in terms
of their Young's moduli, shear moduli, and Poisson ratios. Finally,
we derive a reformulation of the elasticity system that is valid for
both constrained and unconstrained materials and involves only one
scalar unknown in addition to the displacements. For a two-dimensional
constrained material a further reduction to a single scalar equation is
outlined.

We prove apriori estimates and continuous dependence on the
elastic moduli for the equations of homogeneous orthotropic elasticity.
These results are uniform with respect to the three Poisson rations,
Young's moduli, and shear moduli of the material for certain ranges of
these constants. These ranges include the possibility that the compliance
tensor is singular such as occurs for incompressible materials.

Most boundary element methods for two-dimensional boundary
value problems are based on point collocation on the boundary and
the use of splines as trial functions. Here we present a unified
asymptotic error analysis for even as well as for odd degree splines
subordinate to uniform or smoothly graded meshes and prove asymptotic
convergence of optimal order. The equations are collocated at the
breakpoints for odd degree and the internodal midpoints for even degree
splines. The crucial assumption for the generalized boundary integral
and integro-differential operators is strong ellipticity. Our analysis
is based on simple Fourier expansions. In particular, we extend results
by J. Saranen and W.L. Wendland from constant to variable coefficient
equations. Our results include the first convergence proof of midpoint
collocation with piecewise constant functions, i.e., the panel method
for solving systems of Cauchy singular integral equations.

We discuss a technique of implementing certain mixed finite
elements based on the use of Lagrange multipliers to impose
interelement continuity. The matrices arising from this implementation
are positive definite. Considering some well-known mixed methods,
namely the Raviart-Thomas methods for second order elliptic problems
and the Hellan-Hermann-Johnson method for biharmonic problems, we show
that the computed Lagrange multipliers may be exploited in a simple
postprocess to produce better approximation of the original variables.
We further extablish an equivalence between the mixed methods and
certain modified versions of well-known nonconforming methods, notably
the Morley method in the case of the biharmonic problem. The
equivalence is exploited to provide error estimates for both the mixed
and nonconforming methods.

We present in this paper a new velocity-pressure finite
element for the computation of Stokes flow. We discretize the velocity
field iwth continuous piecewise linear functions enriched by bubble
functions, and the pressure by piecewise linear functions. We show
that this element satisfies the usual inf-sup condition and converges
with first order for both velocities and pressures. Finally we relate
this element to families of higher order elements and to the popular
Taylor-Hood element.

A mixed formulation for boundary value problems in
linear elastostatics is presented. This formulation differs slightly
from the classical Hellinger-Reissner formulation. The unknown fields
are the displacement and a tensor related but not equal to the stress.
The tensors appearing in the formulation need not be symmetric, and
consequently mixed finite elements developed for scalar second order
elliptic problems may be applied directly.

We investigate the asymptotic convergence properties of a
variety of methods for the numerical solution of the system of singular
integral equations arising from the traction problem of plane elasticity.
Various sorts of Galerkin methods and collocation methods are considered,
all of which determine a spline approximation via paring with certain test
functions; the test functions may be splines of the same degree as the
trial functions (ordinary Galerkin methods), splines of different degree
(Petrov-Galerkin methods), delta functions (collocation), or trigonometric
polynomials (spline-trig methods). The choice of test functions is
shown to have a significant influence on the convergence properties.

A mixed finite element procedure for plane elasticity is
introduced and analyzed. The symmetry of the stress tensor is enforced
through the introduction of a Lagrange multiplier. An additional
Lagrange multiplier is instroduced to simplify the algebraic system.
Applications are made to incompressible elastic problems and to
plasticity problems.

The Dirichler problem for the equations of plane elasticity
is approximated by a mixed finite element method using a new family
of composite finite elements having properties analogous to those
possessed by the Raviart-Thomas mixed finite elements for a scalar,
second-order elliptic equation. Estimates of optimal order and minimal
regularity are derived for the errors in the displacement vector and
the stress tensor in L^2 and optimal order negative norm estimates are
obtained in (H^s)' for a range of s depending on the index of the finite
element space. An optimal order estimate inL in L_infinity for the
displacement error is given. Also, a quasioptimal estimate is derived in
an appropriate space. All estimates are valid uniformly with respect to
the compressibility and apply in the incompressible case. The formulation
of the elements is presented in detail.

We examine the asymptotic accuracy of the method of
collocation for the approximate solution of linear elliptic partial
differential equations. Specifically we consider the nodal collocation
of a second order equation in the plane with biperiodicity conditions
using tensor product smooth splines of odd degree as trial functions. We
prove optimal rates of convergence in L2 for partial derivatives of the
approximate solution which are of order at least two in one variable,
while the solution itself and its gradient converge in L2 at rates less
than the optimal approximation theoretic results.

Principles for the selection of a finite element method
for a particular problem are discusses. These principles are stated
in terms of the notion of approximability, optimality, and stability.
Several examples are discussed in details as illustrations. Conclusions
regarding the selection of finite element methods are summarized in the
final section of the paper.

We prove quasioptimal and optimal order estimates in various
Sobolev norms for the approximation of linear strongly elliptic
pseudodifferential equations in one independent variable by the method
of nodal collocation by odd degree polynomial splines. The analysis
pertains in particular to many of the boundary element methods used for
numerical computation in engineering applications. Equations to which
the analysis is applied include Fredholm integral equations of the
second kind, certain first kind Fredholm equations, singular integral
equations involving Cauchy kernels, a variety of integro-differential
equations, and two-point boundary value problems for ordinary
differential equations. The error analysis is based on an equivalence
which we establish between the collocation methods and certain
nonstandard Galerkin methods. We compare the collocation method with a
standard Galerkin method using splines of the same degree, showing that
the Galerkin method is quasioptimal in a Sobolev space of lower index
and furnishes optimal order approximation for a range of Sobolev indices
containing and extending below that for the collocation method, and so
the standard Galerkin method achieves higher rates of convergence.

We consider a Galerkin method for functional equations in one
space variable which uses periodic cardinal splines as trial functions
and trigonometric polynomials as test functions. We analyze the method
applied to the integral equation of the first kind arising from a
single layer potential formulation of the Dirichlet problem in the
interior or exterior of an analytic plane curve. In contrast to ordinary
spline Galerkin methods, we show that the method is stable, and so
provides quasioptimal approximation, in a large family of Hilbert spaces
including all the Sobolev spaces of negative order. As a consequence we
prove that the approximate solution to the Dirichlet problem and all its
derivatives converge pointwise with exponential rate.

The goal of engineering computations is to obtain
quantitative information about engineering problems. This goal is
usually achieved by the approximation solution of a mathematically
formulated problem. Although a relevant mathematical formulation of the
problem and its approximation solution are closely related, here we shall
suppose that a mathematical formulation has already been determined and is
amenable to an approximate treatment. We shall discuss a broad class of
approaches based on variational methods of discretization which allow one
to find the approximation solution within a desired range of accuracy.
We discuss properties of these methods which enable us to distinguish
among them and which aid in the selection or design of a method which
is effective in achieving the given goals of the computation.

We compare the efficiency of the solution of
two-dimensional elliptic boundary value problems via boundary integral
methods using two different discretization procedures with comparable
convergence rates: Galerkin procedures with numerical integration and
collocation.

A new semidiscrete finite element method for the solution of
second order nonlinear parabolic boundary value problems is formulated
and analyzed. The test and trial spaces consist of discontinuous
piecewise polynomial functions over quite general meshes with
interelement continuity enforced approximately by means of penalties.
Optimal order error estimates in energy and L2-norms are stated in terms
of locally expressed quantities. They are proved first for a model
problem and then in general.

An unconditionally stable fully discrete finite element
method for the Korteweg-de Vries equation is presented. In addition to
satisfying optimal order global estimates, it is shown that this method
is superconvergent at the nodes. The algorithm is derived from the
conservative method proposed by the second author by the introduction of
a small time-independent forcing term into the discrete equations. This
term is a form of the quasiprojection which was first employed in the
analysis of superconvergence phenomena for parabolic problems. However,
in the present work, unlike in the parabolic case, the quasiprojection
is used as perturbation of the discrete equations and does not affect
the choice of initial values.

The discretization by finite elements of a model variational
problem for a clamped loaded beam is studied with emphasis on the effect
of the beam thickness, which appears as a parameter in the problem, on
the accuracy. It is shown that the approximation achieved by a standard
finite element method degenerates for thin beams. In contrast a large
family of mixed finite element methods are shown to yield quasioptimal
approximation independent of the thickness parameter. The most useful
of these methods may be realized by replacing the integrals appearing
in the stiffness matrix of the standard method by Gauss quadratures.

A convergence analysis is presented for standard and
mixed finite element discretizations of a model system of equations for
a transversely loaded beam. The equations depend parametrically on the
beam thickness and the emphasis of the analysis is on the robustness of
the methods with respect to this parameter. The mixed methods are shown
to be far more robust than the standard methods employing elements of the
same degree. Moveover they entail no additional computational expense.
Computational results are included to illustrate the main results.

A standard Galerkin method for a quasilinear equation of
Sobolev type using continuous, piecewise-polynomial spaces is presented
and analyzed. Optimal order error estimates are established in various
norms, and nodal superconvergence is demonstrated. Discretization in
time by explicit single-step methods is discussed.

The asympotic expansion of the Galerkin solution of a parabolic equation by
means of a sequence of elliptic projections that was introduced by Douglas, Dupont, and
Wheeler is carried out for a quasilinear equation. This quasi-projection can be applied
to establish knot superconvergence in the case of a single space variable. In addition,
an optimal order error estimate in L-infinity(L-infinity) is derived for a single space
variable.